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| [[Image:Ruled hyperboloid.jpg|thumb|right|A [[hyperboloid]] of one sheet is a doubly ruled surface: it can be generated by either of two families of straight lines.]]
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| In [[geometry]], a [[surface]] ''S'' is '''ruled''' (also called a '''scroll''') if through every point of ''S'' there is a straight line that lies on ''S''. The most familiar examples (illustrated here in three-dimensional [[Euclidean space]]) are the [[plane (mathematics)|plane]] and the curved surface of a [[cylinder (geometry)|cylinder]] or [[cone (geometry)|cone]]. Other examples are a [[conical surface]] with [[ellipse|elliptical]] [[directrix (rational normal scroll)|directrix]], the [[right conoid]], the [[helicoid]], and the [[tangent developable]] of a smooth [[curve]] in space.
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| A ruled surface can always be described (at least locally) as the set of points swept by a moving straight line. For example, a cone is formed by keeping one point of a line fixed whilst moving another point along a [[circle]].
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| A surface is '''doubly ruled''' if through every one of its points there are ''two'' distinct lines that lie on the surface. The [[hyperbolic paraboloid]] and the [[hyperboloid of one sheet]] are doubly ruled surfaces. The plane is the only surface which contains at least three distinct lines through each of its points.
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| The properties of being ruled or doubly ruled are preserved by [[projective map]]s, and therefore are concepts of [[projective geometry]]. In algebraic geometry ruled surfaces are sometimes considered to be surfaces in affine or projective space over a field, but they are also sometimes considered as abstract algebraic surfaces without an embedding into affine or projective space, in which case "straight line" is understood to mean an affine or projective line.
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| ==Ruled surfaces in differential geometry==
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| ===Parametric representation===
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| [[Image:Helicoid.svg|thumb|350px|right|A ruled [[helicoid]]]]
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| The "moving line" view means that a ruled surface has a [[parametric equation|parametric representation]] of the form
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| :<math>S(t,u) = p(t) + u r(t)\ </math>
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| where <math>S(t,u)</math> is the generic point on the surface, <math>p(t)</math> is point that traces a curve lying on the surface, and <math>r(t)</math> is a [[unit vector|unit-length vector]] that traces a curve on the [[unit sphere]]. Thus, for example, if one uses
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| :<math>
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| \begin{align}
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| p(t) &= (\cos(2t), \sin(2t), 0)\\
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| r(t) &= ( \cos t \cos 2 t , \cos t \sin 2 t, \sin t )
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| \end{align}
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| </math>
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| one obtains a ruled surface that contains the [[Möbius strip]].
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| Alternatively, a ruled surface can be [[parametric model|parametrized]] as <math>S(t,u) = (1-u) p(t) + u q(t)</math>, where <math>p</math> and <math>q</math> are two non-intersecting curves lying on the surface. In particular, when <math>p(t)</math> and <math>q(t)</math> move with constant speed along two [[skew lines]], the surface is a [[hyperbolic paraboloid]], or a piece of an hyperboloid of one sheet.
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| ===Developable surface===
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| {{Main|Developable surface}}
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| A [[developable surface]] is a surface that can be (locally) unrolled onto a flat plane without tearing or stretching it. If a developable surface lies in three-dimensional Euclidean space, and is [[complete space|complete]], then it is necessarily ruled, but the converse is not always true. For instance, the cylinder and cone are developable, but the general hyperboloid of one sheet is not. More generally, any developable surface in three dimensions is part of a complete ruled surface, and so itself must be locally ruled. There are developable surfaces embedded in four dimensions which are however not ruled. {{harv|Hilbert|Cohn-Vossen|1952|pp=341–342}}
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| ==Ruled surfaces in algebraic geometry==
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| [[Image:hyperbolic-paraboloid.svg|thumb|250px|left|A doubly ruled [[hyperbolic paraboloid]] with equation ''z''=''xy'']]
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| In algebraic geometry, ruled surfaces were originally defined as [[projective surface]]s in [[projective space]] containing a straight line through any given point. This immediately implies that there is a projective line on the surface through any given point, and this condition is now often used as the definition of a ruled surface: ruled surfaces are defined to be abstract projective surfaces satisfying this condition that there is a projective line though any point. This is equivalent to saying that they are [[birational]] to the product of a curve and a projective line. Sometimes a ruled surface is defined to be one satisfying the stronger condition that it has a [[fibration]] over a curve with fibers that are projective lines. This excludes the projective plane, which has a projective line though every point but cannot be written as such a fibration.
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| Ruled surfaces appear in the [[Enriques classification]] of projective complex surfaces, because every algebraic surface of [[Kodaira dimension]] −∞ is a ruled surface (or a projective plane, if one uses the restrictive definition of ruled surface).
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| Every minimal projective ruled surface other than the projective plane is the projective bundle of a 2-dimensional vector bundle over some curve. The ruled surfaces with base curve of genus 0 are the [[Hirzebruch surface]]s.
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| ==Ruled surfaces in architecture==
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| Doubly ruled surfaces are the inspiration for curved [[hyperboloid structure]]s that can be built with a [[latticework]] of straight elements, namely:
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| * Hyperbolic paraboloids, such as [[saddle roof]]s.
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| * Hyperboloids of one sheet, such as [[cooling tower]]s and some [[waste container|trash bin]]s.
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| The [[RM-81 Agena]] [[rocket engine]] employed straight [[cooling channel]]s that were laid out in a ruled surface to form the throat of the [[nozzle]] section.
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| <gallery> | |
| Image:Escuelas Sagrada Familia.jpg|The roof of the school at [[Sagrada Familia]] is a sinusoidally ruled surface.
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| Image:Didcot power station cooling tower zootalures.jpg|Cooling [[Hyperboloid structure|hyperbolic towers]] at [[Didcot Power Station]], UK; the surface can be doubly ruled.
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| Image:Ciechanow water tower.jpg|Doubly ruled water tower with [[Toroid (geometry)|toroidal]] tank, by Jan Bogusławski in [[Ciechanów]], Poland
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| Image:Kobe port tower11s3200.jpg|A hyperboloid [[Kobe Port Tower]], [[Kobe]], Japan, with a double ruling.
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| Image:Shukhov tower shabolovka moscow 02.jpg|The [[gridshell]] of [[Shukhov Tower]] in Moscow, whose sections are doubly ruled.
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| Image:Cremona, torrazzo interno 02 scala a chiocciola.JPG|A ruled helicoid spiral staircase inside [[Cremona]]'s [[Torrazzo of Cremona|Torrazzo]].
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| Image:Nagytotlak.JPG|Village church in Selo, Slovenia: both the roof and the wall are ruled surfaces.
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| Image:W-wa Ochota PKP-WKD.jpg|A [[paraboloid|hyperbolic paraboloid]] roof of [[Warszawa Ochota railway station]] in [[Warsaw]], Poland.
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| Image:Aodai-nonla-crop.jpg|A ruled [[Pointed hat|conical hat]].
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| </gallery> | |
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| ==See also==
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| * [[Differential geometry of surfaces#Ruled surfaces|Differential geometry of ruled surfaces]]
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| * [[Conoid]]
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| * [[Helicoid]]
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| * [[Rational normal scroll]], ruled surface built from two [[rational normal curve]]s
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| ==References==
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| *{{Citation | last1=Barth | first1=Wolf P. | last2=Hulek | first2=Klaus | last3=Peters | first3=Chris A.M. | last4=Van de Ven | first4=Antonius | title=Compact Complex Surfaces | publisher= Springer-Verlag, Berlin | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. | isbn=978-3-540-00832-3 | id={{MathSciNet | id = 2030225}} | year=2004 | volume=4}}
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| *{{Citation | last1=Beauville | first1=Arnaud | title=Complex algebraic surfaces | publisher=[[Cambridge University Press]] | edition=2nd | series=London Mathematical Society Student Texts | isbn=978-0-521-49510-3 | id={{MathSciNet | id = 1406314}} | year=1996 | volume=34}}
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| *{{citation|first=John|last= Sharp
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| |title=D-Forms|publisher=Tarquin|year= 2008}}. Models exploring rules surfaces Review: Jrnl of Mathematics and the Arts 3 (2009), 229-230 ISBN 978-1-899618-87-3
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| *{{citation|first=W. L.|last= Edge
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| |title=The Theory of Ruled Surfaces
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| |publisher=Cambridge, University Press|year= 1931}}. Review: Bull. Amer. Math. Soc. 37 (1931), 791-793, {{doi|10.1090/S0002-9904-1931-05248-4}}
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| * {{Citation | last1=Hilbert | first1=David | author1-link=David Hilbert | last2=Cohn-Vossen | first2=Stephan | author2-link=Stephan Cohn-Vossen | title=Geometry and the Imagination | publisher=Chelsea | location=New York | edition=2nd | isbn=978-0-8284-1087-8 | year=1952 }}.
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| *{{SpringerEOM |id=R/r082790 |title=Ruled surface |first=V.A. |last=Iskovskikh}}
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| ==External links==
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| * {{MathWorld |title=Ruled Surface |id=RuledSurface}}
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| * [http://math.arizona.edu/~models/Ruled_Surfaces Ruled surface pictures from the University of Arizona]
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| * [http://www.rhino3.de/design/modeling/developable/ Examples of developable surfaces on the Rhino3DE website]
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| [[Category:Surfaces]]
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| [[Category:Differential geometry]]
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| [[Category:Differential geometry of surfaces]]
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| [[Category:Complex surfaces]]
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| [[Category:Algebraic surfaces]]
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| [[Category:Geometric shapes]]
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| [[Category:Analytic geometry]]
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43 year old Media Manufacturer (Excluding Video ) Arron from Tisdale, likes to spend some time frolf (frisbee golf), get rid of herpes and operating on cars. Enjoys travel and ended up encouraged after going to Škocjan Caves.
Here is my website - herpes virus cure